Question

Use Stokes" Theorem to evaluate (F-dr where F(x, y, z)=(-y , x-z , 0) and the surface S is the part of the paraboloid : z = 4- x^2 - y^2 that lies above the xy-plane. Assume C is oriented counterclockwise when viewed from above.

Answer #1

Use Stokes' Theorem to evaluate
S
curl F · dS.
F(x, y, z) = x2 sin(z)i + y2j + xyk,
S is the part of the paraboloid
z = 1 − x2 − y2
that lies above the xy-plane, oriented upward.

Use Stokes' Theorem to evaluate
∫
C
F · dr
where F = (x +
8z) i + (6x +
y) j + (7y −
z) k and C is the curve of
intersection of the plane x + 3y + z
= 24 with the coordinate planes.
(Assume that C is oriented counterclockwise as viewed from
above.) Please explain steps. Thank you:)

Use Stokes' Theorem to evaluate
C
F · dr
where C is oriented counterclockwise as viewed from
above.
F(x, y, z) = 5yi + xzj + (x + y)k,
C is the curve of intersection of the plane
z = y + 7
and the cylinder
x2 + y2 = 1.

Use Stokes' Theorem to evaluate
C
F · dr
where C is oriented counterclockwise as viewed from
above.
F(x, y,
z) = yzi +
6xzj +
exyk,
C is the circle
x2 +
y2 = 9, z = 2.

Use Stokes' Theorem to evaluate ∫ C F ⋅ dr. In each case C is
oriented counterclockwise as viewed from above. F ( x , y , z ) = e
− x ˆ i + e x ˆ j + e z ˆ k
C is the boundary of the part of the plane 2 x + y + 2 z = 2 in
the first octant ∫ C F ⋅ d r =

Use Stokes' Theorem to evaluate ∫ C F · dr where F = (x +
5z) i + (3x + y) j + (4y − z) k and C is the curve of
intersection of the plane x + 2y + z = 16 with the coordinate
planes

Evaluate F · dr, where F(x, y) = <(xy), (3y^2)> and C is
the portion of the circle x^2 + y^2 = 4 from (0, 2) to (0, −2)
oriented counterclockwise in the xy-plane.

Evaluate∫ C F ⋅ d r , where F(x,y,z)={e^−4x,e^2x,1e^z} and C is
the boundary of the part of the plane 3x+7y+5z=3 lying in the first
octant, traversed counterclockwise as viewed from above. HINT: Use
Stokes' Theorem.

Evaluate H C F · dr, if F(x, y, z) = yi + 2xj + yzk, and C is
the curve of intersection of the part of the paraboliod z = 1 − x 2
− y 2 in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with the coordinate
planes x = 0, y = 0 and z = 0, oriented counterclockwise when
viewed from above. The answer is pi/4+4/15

Use Stokes' Theorem to evaluate the integral
∮CF⋅dr=∮C8z^2dx+8xdy+2y^3dz where C is the circle x^2+y^2=9 in the
plane z=0 .

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